There is no way through the REST API to move a message.
This specific question was asked in this post: http://lithosphere.lithium.com/t5/developers-discussion/How-to-move-posts-using-REST-API/m-p/51838/highlight/true#M1732 .
If some kind of custom move message flow needs to be developed, I would recommend reaching our to our Services group with your specific use-case -- they may be able to provide a solution that meets the requirements.
I'm not sure if you'd want to duplicate posts in other boards (I guess it depends on your use-case, and it sounds like in this use-case maybe you do), but the API does allow you to post new message, edit an existing message, and reply to a message. We do not allow you to delete messages through our API (which doesn't seem to be a need in this case).